%% slac-pub-7176: page file slac-pub-7176-0-0-2.tcx.
%% section 2 IR sensitivity of LLA and soft gluon emission at large angles [slac-pub-7176-0-0-2 in slac-pub-7176-0-0-2: slac-pub-7176-0-0-3]
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\section{\usemenu{slac-pub-7176::context::slac-pub-7176-0-0-2}{IR sensitivity of LLA and soft gluon emission at large angles}}\label{section::slac-pub-7176-0-0-2}
We provide evidence \cite{1,4,5} that the Drell-Yan cross section
is free from $1/Q$ IR contributions.
To one-loop accuracy this statement can be checked by an explicit
calculation with a small gluon mass $\lambda$ as regulator.
Nonanalytic terms in the small-$\lambda$
expansion correspond to higher twist contributions \cite{6,7}.
A textbook calculation gives \cite{1}
at $N\gg 1$
\be
\omega_{DY}(N,Q^2,\lambda^2)-\omega_{DY}(N,Q^2,0) =
\frac{C_F \alpha_s}{\pi}\frac{N^2\lambda^2}{Q^2}\ln(\lambda^2/Q^2)+
O\!\left(\frac{1}{N},\frac{N\lambda}{Q} \right).
\ee
Note that the suspected linear term $\sqrt{\lambda^2/Q^2}$ is absent; the
leading IR contribution is of order $N^2\lambda^2/Q^2$. The $N^2$
enhancement signals that the power corrections are determined by the
{\em smaller} of the two large scales: $Q/N$ is the
moment space analogue of the energy available for gluon emission.
In terms of the energy fraction $z$ the relevant scale is $(1-z)Q$ rather
the mass of the Drell-Yan pair $Q$.
To understand the apparent $1/Q$ sensitivity of the LLA, it is
instructive to consider the structure of the
one-loop integral for soft gluon emission
\be
\omega_{DY} \sim \int\frac{d^3k}{2k_0}\delta[(p_1+p_2-k)^2-Q^2]
\left|{\cal M_{DY}}\right|^2.
\ee
The matrix element (in LLA) is $\left|{\cal M_{DY}}\right|^2
\sim 2Q^2/k_\perp^2$
and it is convenient to rewrite the phase-space integral as
$$
\int \frac{d^3k}{2k_0} \sim \int\frac{dk_\perp^2}{\sqrt{k_0^2-k_\perp^2}}\,.
$$
We now take a massless gluon, and to avoid collinear divergences
introduce an explicit cutoff on the minimal transverse momentum
$k_\perp>\mu$. Remembering that $k_0=\sqrt{s}(1-z)/2$ and taking moments
we get
\be\label{DYphasespace}
\omega_{DY}^{soft} = \frac{2C_F}{\pi}\int_0^{1-2\mu/Q} dz\, z^{N-1}
\int_{\mu^2}^{Q^2(1-z)^2/4}\frac{d k^2_\perp}{k^2_\perp}\frac{1}
{\sqrt{(1-z)^2-4k^2_\perp/Q^2}}
\ee
The crucial point is now that the LLA corresponds to resummation of
soft and {\em collinear} emission, that is the leading logarithms
(and in fact the next-to-leading as well) come from the integration
region of small gluon transverse momentum compared to its energy
$k_\perp \ll k_0 = Q(1-z)/2\ll Q$. Thus, for summation of
large logarithms, it is safe to neglect the term $4 k_\perp^2/Q^2$
under the square root, so that
\be
\omega_{DY}^{soft+coll.} = \frac{2C_F}{\pi}\int_0^{1-2\mu/Q} dz\,
\frac{ z^{N-1}-1}{1-z}
\int_{\mu^2}^{Q^2(1-z)^2/4}\frac{d k^2_\perp}{k^2_\perp},
\ee
where we have replaced $z^{N-1}$ by $z^{N-1}-1$ to
take into account virtual gluon exchange.
Taking the integrals, we get the expected double logarithm but
also the linear term in $\mu/Q$ discussed in Sect.~1.
However, this IR contribution of order $1/Q$ comes from the
end-point integration region $1-z\sim \mu/Q$ (where $k_\perp\sim k_0$)
in which neglect of
the $k_\perp^2/Q^2$ term under the square root is not justified.
In fact, the square root cannot even be expanded in $k_\perp^2/
(Q^2 (1-z)^2)$
since this would generate increasingly singular contributions. Instead, the
integral must be taken exactly. When this is done \cite{1}, all
$\mu/Q$ terms disappear.
The physical reason for the enhanced IR sensitivity of the resummed
cross section in LLA is that we neglected soft gluon radiation
at large angles $k_\perp\sim k_0$. To recover the correct
IR behavior, the phase space integral for soft gluon
emission has to be taken exactly; the common collinear approximation is
sufficient for summing logarithms to leading and next-to-leading
logarithmic accuracy
but is misleading for the analysis of power-suppressed effects.
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